Finite Group Databasebeta

← C2×#10613190 #12971435

Subgroups — C2×#10613190 #12971435

Order 1280.

2018 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65958C211yesmaximal, characteristic
2C22×#4034C211yesmaximal, characteristic, complemented
3C2×C5×#484C211yesmaximal, characteristic, complemented
4C2×#65333C211yesmaximal, characteristic, complemented
5C2×#65257C211yesmaximal, characteristic
6#10613190C2116yesmaximal, direct product factor
7C2×#65179C212yesmaximal, complemented
8C2×#65165C212yesmaximal, complemented
9C2×#65084C212yesmaximal, complemented
10C2×#65080C212yesmaximal, complemented
11C2×#65024C211yesmaximal, characteristic
12C2×#65008C211yesmaximal, characteristic, complemented
13C23×Dic10—21no—
14C2×C5×C22⋊Q8C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C2×#4093C2211yescharacteristic
17C2×Q8×Dic5C2211yescharacteristic
18C2×#4076C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C22×#683—21no—
21C22×C4×Dic5C2211yescharacteristic
22#65958—24no—
23#65958C2218yes—
24C2×#4044—21no—
25C2×#4039C2211yescharacteristic
26C2×#4038—21no—
27C2×#4036C2211yescharacteristic
28C2×#4034—21no—
29C2×#4034—21no—
30C2×#4034—24no—
31C2×#4034C2212yes—
32C2×#4034C2212yes—
33C2×#4034C2218yescomplemented
34C2×#4032—21no—
35C2×#4027—21no—
36C5×#484C2218yescomplemented
37C2×C5×#107C2212yes—
38C2×C5×#103C2212yes—
39C2×C5×C22⋊C8C2211yescharacteristic
40C2×#3967C2212yes—
41C2×C22⋊#168C2211yescharacteristic
42C2×#3953C2212yes—
43C2×#3949C2212yes—
44C2×#3947C2212yes—
45C2×#3946C2212yes—
46#65333C2218yescomplemented
47#65257C2218yes—
48C2×#3941C2212yes—
49C2×#3940C2212yes—
50C2×#3939C2212yes—
51#65179C22116yescomplemented
52#65165C22116yescomplemented
53#65084C22116yescomplemented
54#65080C22116yescomplemented
55#65024C2218yes—
56#65008C2218yescomplemented
57C2×#3239—51nomaximal, Sylow
58C22×C5×Q8C2311yescharacteristic
59C23×C4×C5—21no—
60C23×C4×C5C2311yescharacteristic
61C22×C22⋊D5—21no—
62C23×Dic5—21no—
63C23×Dic5D411yescharacteristic
64C23×Dic5D411yescharacteristic
65C22×D20C2311yescharacteristic
66C22×C4×D5—21no—
67C22×Dic10—22no—
68C22×Dic10—22no—
69C22×Dic10—28no—
70C22×Dic10—48no—
71C22×C5×C8C2312yes—
72C5×C22⋊Q8—22no—
73C5×C22⋊Q8C2314yes—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—22no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4C2312yes—
78C2×C5×C4⋊C4C2314yes—
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C22⋊C4—21no—
81#4093—22no—
82#4093C2314yes—
83Q8×Dic5—22no—
84Q8×Dic5C2314yes—
85#4076—22no—
86#4076C2314yes—
87C2×#685—21no—
88C2×#685—21no—
89C2×#685—21no—
90C2×#685—21no—
91C2×C4⋊Dic5—21no—
92C2×C4⋊Dic5—22no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5C2314yes—
96C2×C4⋊Dic5C2312yes—
97C2×C4⋊Dic5C2311yescharacteristic
98C2×#683—21no—
99C2×#683—22no—
100C2×#683—22no—
101C2×#683—28no—
102C2×C4×Dic5—21no—
103C2×C4×Dic5D411yescharacteristic
104C2×C4×Dic5D411yescharacteristic
105C2×C4×Dic5D414yes—
106C2×C4×Dic5C2311yescharacteristic
107C2×C4×Dic5C2311yescharacteristic
108C2×C4×Dic5C2314yes—
109C22×#168C2312yes—
110#4044—28no—
111#4039—22no—
112#4039C2314yes—
113#4038—28no—
114#4036—22no—
115#4036C2314yes—
116#4034—24no—
117#4034—24no—
118#4034—28no—
119#4034—28no—
120#4034C2318yes—
121#4034C2318yes—
122#4032—28no—
123#4027—28no—
124C5×#107C2318yes—
125C5×#103C2318yes—
126C5×C22⋊C8C2314yes—
127#3967C2318yes—
128C22⋊#168C2314yes—
129#3953C2318yes—
130#3949C2318yes—
131#3947C2318yes—
132#3946C2318yes—
133#3941C2318yes—
134#3940C2318yes—
135#3939C2318yes—
136C22×C4⋊Q8—51no—
137C2×#529—51no—
138#3239—516no—
139C2×#501—52no—
140C2×#489—52no—
141C2×#484—51no—
142C2×#484D511yescharacteristic, complemented
143C2×#481—51no—
144C2×#481—51no—
145C2×#475—52no—
146C2×#474—52no—
147C2×#432—51no—
148C24×C5C2×D411yescharacteristic
149C23×D5—21no—
150C2×C5×Q8—21no—
151C2×C5×Q8—24no—
152C2×C5×Q8C2412yes—
153C22×C4×C5—24no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—24no—
157C22×C4×C5—22no—
158C22×C4×C5—44no—
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D412yes—
161C22×C4×C5C2412yes—
162C22×C4×C5C2411yescharacteristic
163C2×C22⋊D5—24no—
164C2×C22⋊D5—44no—
165C22×Dic5—24no—
166C22×Dic5—24no—
167C22×Dic5—24no—
168C22×Dic5—22no—
169C22×Dic5—21no—
170C22×Dic5—44no—
171C22×Dic5C2×D414yes—
172C22×Dic5C2×D411yescharacteristic
173C22×Dic5C2×D414yes—
174C22×Dic5C2×D411yescharacteristic
175C22×Dic5C2×D411yescharacteristic
176C22×Dic5C2×D411yescharacteristic
177C2×D20—24no—
178C2×D20—21no—
179C2×D20C2412yes—
180C2×C4×D5—24no—
181C2×C4×D5—44no—
182C2×Dic10—28no—
183C2×Dic10—28no—
184C2×Dic10—48no—
185C2×Dic10—432no—
186C2×Dic10—48no—
187C2×C5×C8—24no—
188C2×C5×C8C2414yes—
189C5×C4⋊C4—24no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—24no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4C2414yes—
194C5×C4⋊C4C2412yes—
195C5×C22⋊C4—24no—
196#685—24no—
197#685—24no—
198#685—24no—
199#685—24no—
200C4⋊Dic5—24no—
201C4⋊Dic5—22no—
202C4⋊Dic5—21no—
203C4⋊Dic5—24no—
204C4⋊Dic5—24no—
205C4⋊Dic5C2414yes—
206C4⋊Dic5C2412yes—
207#683—24no—
208#683—28no—
209#683—28no—
210C4×Dic5—24no—
211C4×Dic5C2×D412yes—
212C4×Dic5C2×D414yes—
213C4×Dic5C2×D412yes—
214C4×Dic5C2×D414yes—
215C4×Dic5C2412yes—
216C4×Dic5C2412yes—
217C2×#168—24no—
218C2×#168C2414yes—
219C23×Q8—101no—
220C2×C4⋊Q8—52no—
221C2×C4⋊Q8—52no—
222C2×C4⋊Q8—58no—
223C2×C4⋊Q8—101no—
224C2×C4⋊Q8—101no—
225C2×C4⋊Q8—104no—
226#529—58no—
227#529—104no—
228C2×#126—101no—
229C2×#125—51no—
230C2×#124—51no—
231C2×#123—101no—
232C2×C22⋊Q8C2×D511yescharacteristic
233C2×#121—51no—
234C2×C4×Q8—51no—
235C2×C4×D4—51no—
236C2×C4×D4—101no—
237C2×#117—101no—
238C22×C4⋊C4—51no—
239C22×C4⋊C4—101no—
240C22×C4⋊C4C2×D511yescharacteristic
241C22×C42—51no—
242#501—516no—
243#489—516no—
244#484—58no—
245#484C2×D518yescomplemented
246#481—58no—
247#481—58no—
248#475—516no—
249#474—516no—
250#432—58no—
251C2×#107—52no—
252C2×#107C2×D512yes—
253C2×#106—52no—
254C2×#106—52no—
255C2×C4⋊C8—52no—
256C2×#103—52no—
257C2×#103C2×D512yes—
258C2×#102—52no—
259C2×#102—52no—
260C2×C22⋊C8—51no—
261C2×C22⋊C8C2×D511yescharacteristic
262C2×#97—52no—
263C23×C5—22no—
264C23×C5—22no—
265C23×C5#13714yes—
266C23×C5C22×D412yes—
267C23×C5C22×D411yescharacteristic
268C22×D5—22no—
269C22×D5—42no—
270C22×D5—42no—
271C22×D5—42no—
272C5×Q8—24no—
273C5×Q8—42no—
274C2×C4×C5—22no—
275C2×C4×C5—24no—
276C2×C4×C5—24no—
277C2×C4×C5—22no—
278C2×C4×C5—24no—
279C2×C4×C5—22no—
280C2×C4×C5—48no—
281C2×C4×C5—42no—
282C2×C4×C5—44no—
283C2×C4×C5—42no—
284C2×C4×C5C22×D412yes—
285C2×C4×C5C22×D411yescharacteristic
286C2×C4×C52−1+414yes—
287C2×C4×C5C2511yescharacteristic
288C22⋊D5—48no—
289C2×Dic5—22no—
290C2×Dic5—28no—
291C2×Dic5—22no—
292C2×Dic5—22no—
293C2×Dic5—28no—
294C2×Dic5—22no—
295C2×Dic5—24no—
296C2×Dic5—48no—
297C2×Dic5—44no—
298C2×Dic5—42no—
299C2×Dic5AGL(1,ℤ/8)14yes—
300C2×Dic5C22×D412yes—
301C2×Dic5C22×D412yes—
302C2×Dic5C22×D412yes—
303C2×Dic5C22×D412yes—
304D20—24no—
305D20—42no—
306C4×D5—48no—
307Dic10—416no—
308Dic10—416no—
309C5×C8—24no—
310#168—24no—
311C22×Q8—108no—
312C22×Q8—102no—
313C22×Q8—102no—
314C22×Q8—208no—
315C22×Q8C22×D511yescharacteristic
316C22×D4—51no—
317C22×D4—101no—
318C23×C4—21no—
319C23×C4—51no—
320C23×C4—51no—
321C23×C4—101no—
322C23×C4—101no—
323C23×C4C22×D511yescharacteristic
324C22×C8—52no—
325C22×C8C22×D512yes—
326C4⋊Q8—58no—
327C4⋊Q8—58no—
328C4⋊Q8—108no—
329C4⋊Q8—104no—
330C4⋊Q8—104no—
331C4⋊Q8—108no—
332#126—108no—
333#125—54no—
334#125—102no—
335#124—54no—
336#124—102no—
337#123—108no—
338C22⋊Q8—22no—
339C22⋊Q8C22×D514yes—
340#121—54no—
341#121—102no—
342C4×Q8—54no—
343C4×Q8—102no—
344C4×D4—54no—
345C4×D4—102no—
346C4×D4—108no—
347#117—108no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—52no—
352C2×C4⋊C4—54no—
353C2×C4⋊C4—51no—
354C2×C4⋊C4—108no—
355C2×C4⋊C4—102no—
356C2×C4⋊C4—102no—
357C2×C4⋊C4—101no—
358C2×C4⋊C4—101no—
359C2×C4⋊C4—101no—
360C2×C4⋊C4—102no—
361C2×C4⋊C4—101no—
362C2×C4⋊C4C22×D512yes—
363C2×C4⋊C4C22×D514yes—
364C2×C4⋊C4C22×D511yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—101no—
367C2×C22⋊C4—101no—
368C2×C22⋊C4—101no—
369C2×C22⋊C4—101no—
370C2×C42—51no—
371C2×C42—51no—
372C2×C42—54no—
373C2×C42—51no—
374C2×C42—51no—
375C2×C42—54no—
376C2×C42—101no—
377#107—58no—
378#107C22×D518yes—
379#106—58no—
380#106—58no—
381C4⋊C8—58no—
382#103—58no—
383#103C22×D518yes—
384#102—58no—
385#102—58no—
386C22⋊C8—54no—
387C22⋊C8C22×D514yes—
388#97—58no—
389C22×C5—22no—
390C22×C5—28no—
391C22×C5—22no—
392C22×C5#53612yes—
393C22×C5C2×AGL(1,ℤ/8)12yes—
394C22×C5C2×#13714yes—
395C22×C5C2×#13712yes—
396C22×C5C23×D411yescharacteristic
397C2×D5—42no—
398C2×D5—42no—
399C2×D5—42no—
400C2×D5—84no—
401C4×C5—22no—
402C4×C5—42no—
403C4×C5—44no—
404C4×C5—42no—
405C4×C5#53612yes—
406C4×C5C2×2−1+412yes—
407Dic5—22no—
408Dic5—22no—
409Dic5—22no—
410Dic5—44no—
411Dic5—42no—
412Dic5C2×AGL(1,ℤ/8)14yes—
413C24—101no—
414C24D4×D511yescharacteristic
415C2×Q8—21no—
416C2×Q8—24no—
417C2×Q8—108no—
418C2×Q8—108no—
419C2×Q8—208no—
420C2×Q8—2032no—
421C2×Q8—208no—
422C2×Q8C23×D512yes—
423C2×D4—52no—
424C2×D4—104no—
425C2×D4—101no—
426C2×D4—104no—
427C2×D4—204no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—51no—
435C22×C4—54no—
436C22×C4—54no—
437C22×C4—51no—
438C22×C4—51no—
439C22×C4—51no—
440C22×C4—104no—
441C22×C4—104no—
442C22×C4—104no—
443C22×C4—104no—
444C22×C4—101no—
445C22×C4—102no—
446C22×C4—204no—
447C22×C4—204no—
448C22×C4D4×D512yes—
449C22×C4D4×D511yescharacteristic
450C22×C4C23×D512yes—
451C22×C4C23×D511yescharacteristic
452C2×C8—24no—
453C2×C8—54no—
454C2×C8—104no—
455C2×C8C23×D514yes—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—54no—
461C4⋊C4—52no—
462C4⋊C4—108no—
463C4⋊C4—108no—
464C4⋊C4—104no—
465C4⋊C4—104no—
466C4⋊C4—104no—
467C4⋊C4—104no—
468C4⋊C4—102no—
469C4⋊C4—101no—
470C4⋊C4C23×D514yes—
471C4⋊C4C23×D512yes—
472C22⋊C4—24no—
473C22⋊C4—104no—
474C22⋊C4—104no—
475C22⋊C4—104no—
476C22⋊C4—104no—
477C42—54no—
478C42—52no—
479C42—52no—
480C42—54no—
481C42—52no—
482C42—52no—
483C42—104no—
484C2×C5—24no—
485C2×C5#323914yescomplemented
486C2×C5C2×#53611yescharacteristic
487C2×C5C22×AGL(1,ℤ/8)11yescharacteristic
488C2×C5C22×#13711yescharacteristic
489D5—82no—
490C23—22no—
491C23—22no—
492C23—102no—
493C23—202no—
494C23—202no—
495C23—202no—
496C23#406414yes—
497C23C2×D4×D512yes—
498C23C2×D4×D511yescharacteristic
499Q8—24no—
500Q8—42no—
501Q8—2016no—
502Q8—2016no—
503D4—104no—
504D4—208no—
505D4—202no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—54no—
517C2×C4—52no—
518C2×C4—52no—
519C2×C4—52no—
520C2×C4—52no—
521C2×C4—108no—
522C2×C4—108no—
523C2×C4—102no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4—104no—
528C2×C4—208no—
529C2×C4—208no—
530C2×C4—202no—
531C2×C4—204no—
532C2×C4C2×D4×D512yes—
533C2×C4C2×D4×D511yescharacteristic
534C2×C4#415014yes—
535C2×C4C24×D511yescharacteristic
536C8—24no—
537C8—104no—
538C5C2×#323911yescharacteristic, complemented, Sylow
539C22—22no—
540C22—28no—
541C22—22no—
542C22—202no—
543C22—202no—
544C22—202no—
545C22—404no—
546C22#6598212yes—
547C22C2×#406414yes—
548C22C2×#406412yes—
549C22D5×AGL(1,ℤ/8)12yes—
550C22C22×D4×D511yescharacteristic
551C4—22no—
552C4—42no—
553C4—44no—
554C4—42no—
555C4—54no—
556C4—102no—
557C4—102no—
558C4—102no—
559C4—202no—
560C4—204no—
561C4#6598212yes—
562C4C2×#415012yes—
563C2—24no—
564C2—402no—
565C2#1061319014yesdirect product factor
566C2C2×#6598211yescharacteristic
567C2C22×#406411yescharacteristic
568C2C2×D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#484C2
socleC23×C5C22×D4
prime radicalC2×#4076C22
hypercenterC2×#484D5
hypocenterC5C2×#3239
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#4034C2semidirect
3. C2×C5×#484C2semidirect
4. C2×#65333C2semidirect
6. #10613190C2direct (N × H)
7. C2×#65179C2semidirect
8. C2×#65165C2semidirect
9. C2×#65084C2semidirect
10. C2×#65080C2semidirect
12. C2×#65008C2semidirect
33. C2×#4034C22semidirect
36. C5×#484C22semidirect
46. #65333C22semidirect
51. #65179C22semidirect
52. #65165C22semidirect
53. #65084C22semidirect
54. #65080C22semidirect
56. #65008C22semidirect
142. C2×#484D5semidirect
245. #484C2×D5semidirect
485. C2×C5#3239semidirect
538. C5C2×#3239semidirect
565. C2#10613190direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#659582
C22×#40342
C2×C5×#4842
C2×#653332
C2×#652572
#106131902
C2×#651792
C2×#651652
C2×#650842
C2×#650802
C2×#650242
C2×#650082
C2×#32395